The paper is organized as follows. In Section 2 we review the basics on Morse-Bott theory, Lie groupoids and Riemannian metrics on Lie groupoids as introduced in [1]}. In Section 3 we introduce the main objects of study of the paper, namely Morse Lie groupoid morphisms. We study their main properties and we show Proposition REF  which establishes that the property of being a Morse Lie groupoid morphism is Morita invariant. In Section 4 we give several examples of Morse Lie groupoid morphisms and we show an existence result for proper Lie groupoids whose base is compact, this is the content of Theorem REF . Then we recall the notion of Hamiltonian actions of Lie 2-groups on 0-symplectic groupoids [2]} and we show that in the case of a Lie 2-group whose base is a torus the components of the moment map are Morse Lie groupoid morphisms. In Section 5 by means of tubular neighborhoods given by Euler-like vector fields [3]}, we show Theorem REF  which is our version of the Morse Lemma in the Lie groupoid setting. In particular, we introduce a notion of index of a non-degenerate critical subgroupoid as well as their positive and negative normal bundles. In Section 6 we study gradient vector fields of real valued Lie groupoid morphisms with respect to a 2-metric in the sense of [1]}. We show Proposition REF  which says the such gradient vector fields are multiplicative. We also prove Proposition REF  which gives an attaching construction of Lie groupoids whose outcome is a topological groupoid. As a consequence, we show Proposition REF  and Theorem REF  which describe critical sub-levels of Morse Lie groupoid morphisms in terms of attaching groupoids. In Section 7 we extend the classical results of Morse-Smale dynamics to the framework of Lie groupoids and we show Proposition REF  which says that the moduli space of gradient flow lines has a natural structure of Lie groupoid. In Section 8 we construct a double complex which is the Lie groupoid analogue of the Morse-Bott complex defined by Austin and Braam in [5]}. The main result of this section is Theorem REF  which shows that the total cohomology of the groupoid double Morse complex is isomorphic to the Bott-Shulman-Stasheff cohomology of the underlying Lie groupoid. In particular, the cohomology of the double groupoid Morse complex is Morita invariant. We finish this section by introducing an equivariant groupoid Morse double complex with respect to an action of a Lie 2-group. We prove Proposition REF  which establishes that the equivariant cohomology of a Lie 2-group action as defined in [6]} can be computed by means of the cohomology of the equivariant groupoid Morse double complex. We use this result to describe the equivariant cohomology of toric symplectic stacks by means of groupoid Morse theoretical tools.
